M1: A Triangle Problem

This is a user suggested problem. The TopsOJ staff thanks and gives full credit to Maximilian113 for their contribution.

Let ABC be a triangle with perimeter 15, and suppose that \displaystyle 16[ABC]^2 = (AB+BC+CA) \cdot AB \cdot BC \cdot CA. If [ABC] is of the form \frac{m\sqrt{p}}{n}, where m and n are relatively prime, positive integers, and p is not divisible by the square of a prime, find m+n+p.